Question

2
12. A variable circle passes through the fixed
point (2,0) and touches y-axis. Then locus of
centre of circle
1) a parabola
2) a circle
3) an ellipse
4) a hyperbola​

Answer 1

Answer:

If the point (2,0) and the point (0,y) are both on the circle as in the question then the centre of the circle must be an equal distance away from each of those two points. Let the centre be (h,k).

So,

sqrt((h-2)^2+(k-0)^2) = sqrt((h-0)^2+(k-y)^2)

h^2-4h+4+k^2=h^2+k^2-2ky+y^2

-4h+4=-2ky+y^2

k=(y^2+4h-4)/2y

When y is given, that is linear. So for any given point on the y-axis where the circle touches, there is a line that the centre of the circle must lie on.